# MIT Probability Seminar

## Current Organizers

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# Spring 2022

Monday 4.15 - 5.15 pm

Room 2-147

Scheduled virtual talks will be held on Zoom, Monday 4:15-5:15 pm.
A link to a Zoom classroom will appear here!!

## Schedule

• February 7

• February 14

Cole Graham (Brown University)

Stationary measures for stochastic conservation laws

Abstract: At long times, many SPDEs relax to statistically steady states. In this talk, I will consider the existence and uniqueness of such stationary measures for stochastically-forced viscous conservation laws on the line. A special case, the stochastic Burgers equation, has received a great deal of attention due to its links to the KPZ and stochastic heat equations. Stochastic Burgers is known to admit a unique spacetime-stationary ergodic measure for each mean. However, existing proofs rely on the Cole–Hopf transformation, which does not extend to other conservation laws. I will discuss a comparison-based approach that recovers partial results for more general conservation laws. In particular, such SPDEs admit infinitely many stationary ergodic measures, and there is at most one such measure for each mean.

This is joint work with Theodore Drivas, Alexander Dunlap, Joonhyun La, and Lenya Ryzhik.

• February 21

• February 28

Hugo Falconet (Courant Institute)

Metric growth dynamics in Liouville quantum gravity.

Abstract: Liouville quantum gravity (LQG) is a canonical model of random geometry. Associated with the planar Gaussian free field, this geometry with special conformal symmetries was introduced in the physics literature by Polyakov in the 80's and is conjectured to describe the scaling limit of random planar maps. In this talk, I will introduce LQG as a metric measure space and discuss recent results on the associated metric growth dynamics. The primary focus will be on the dynamics of the trace of the free field on the boundary of growing LQG balls.

Based on a joint work with Julien Dubédat.

• March 7

Mark Sellke (Stanford University)

Algorithmic Thresholds in Mean-Field Spin Glasses.

Abstract: I will explain recent progress on computing approximate ground states of mean-field spin glass Hamiltonians, which are certain random functions in high dimension. While the asymptotic ground state energy OPT is given by the famous Parisi formula, the landscape of these functions often include many bad local optima which impede optimization by efficient algorithms. In the positive direction, I will explain algorithms based on approximate message passing which asymptotically achieve a value ALG given by an extended Parisi formula. The case ALG=OPT has a "no overlap gap" or "full replica symmetry breaking" interpretation, but in general these algorithms may fail to reach asymptotic optimality. In the negative direction, I will explain why no algorithm with suitably Lipschitz dependence on the random disorder can surpass the threshold ALG. This result applies to many standard optimization algorithms, such as gradient descent and its variants on dimension-free time scales. Based on joint works with Ahmed El Alaoui, Brice Huang, and Andrea Montanari.

• March 14

*** Special Seminar Time 1pm-2pm! Special Location Room 2-132***

Eviatar Procaccia (Technion)

Stationary Hastings-Levitov model.

Abstract: We construct and study a stationary version of the Hastings-Levitov(0) model. We prove that unlike the classical model, in the stationary case, particle sizes are tight, yielding that this model can be seen as a tractable off-lattice Diffusion Limited Aggregation (DLA). The stationary setting together with a geometric interpretations of the harmonic measure yields new geometric results such as finiteness of arms, exact growth rate and fractal dimension equals 3/2, corresponding to a numerical prediction of Meakin from 1983 for the gyration radius of DLA growing on a long line segment. We will also show that similar results can be achieved in a cylinder.

• March 14

Dominik Schmid (Bonn/Princeton)

Mixing times for the TASEP on the circle.

Abstract: The exclusion process is one of the best-studied examples of an interacting particle system. In this talk, we consider simple exclusion processes on finite graphs. We give an overview over some recent results on the mixing time of the totally asymmetric simple exclusion process (TASEP). In particular, we provide bounds on the mixing time of the TASEP on the circle, using a connection to periodic last passage percolation. This talk is based on joint work with Allan Sly (Princeton).

• March 21

Spring Break, no seminar.

• March 28

Oanh Nguyen (Brown University)

Survival time of the contact process on random graphs.

Abstract: The contact process is a model for the spread of infections on graphs. In this talk, we discuss the contact process on random graphs with low infection rate $\lambda$. For random $d$-regular graphs, it is known that the survival time is $O(\log n)$ below the critical $\lambda_c$. By contrast, on the Erdos-Renyi random graphs $G(n,d/n)$, rare high degree vertices result in much longer survival times. We show that the survival time is governed by high density local configurations, in particular large connected components of high degree vertices on which the infection lasts for time $n^{\lambda^{2+o(1)}}$. We shall discuss how to obtain a matching upper bound. Our methods, moreover, generalize to random graphs with given degree distributions that have exponential moments.

• April 4

Johannes Alt (Courant Institute)

Localization and Delocalization in Erdős–Rényi graphs

Abstract: We consider the Erdős–Rényi graph on N vertices with edge probability d/N. It is well known that the structure of this graph changes drastically when d is of order log N. Below this threshold it develops inhomogeneities which lead to the emergence of localized eigenvectors, while the majority of the eigenvectors remains delocalized. In this talk, I will present the phase diagram depicting these localized and delocalized phases and our recent progress in establishing it rigorously.

This is based on joint works with Raphael Ducatez and Antti Knowles.

• April 11

Sky Cao (Stanford University)

Exponential decay of correlations in finite gauge group lattice gauge theories

Abstract: Lattice gauge theories with finite gauge groups are statistical mechanical models, very much akin to the Ising model, but with some twists. In this talk, I will describe how to show exponential decay of correlations for these models at low temperatures. This is based on joint work with Arka Adhikari.

• April 18

• April 25

*** Special Seminar in Room 3-361 at 1pm! ***

Remi Rhodes (Aix-Marseille Université)

Segal’s axioms and conformal bootstrap in Liouville theory.

Abstract:Conformal field theories (CFT) are expected to describe models of statistical physics in 2D undergoing a second order phase transition at their critical point. Several axiomatics have been proposed to lay the mathematical foundations for the concept of CFT. In Segal’s approach, the data for a CFT are an Hilbert space H and a map that associates to each Riemann surface S with boundary a Hilbert-Schmidt operator (called amplitude) acting on the tensor product $H^b$ with b the number of boundary components of S. Amplitudes are then assumed to compose in a natural way under gluing of surfaces along their boundaries. Segal’s approach turned out to be very attractive for mathematicians in view of its geometric flavor. Also, it gives a geometrical way to understand the conformal bootstrap conjecture in physics: correlation functions of CFT should factorize as an integral over their spectrum of the product of (squared) conformal blocks times the structure constants of the CFT (the 3 point correlation functions on the Riemann sphere). Conformal blocks are holomorphic functions of the moduli of the space of Riemann surfaces with marked point, which are universal in the sense that they only depend on the commutation relations of a given Lie algebra, the Virasoro algebra. Structure constants are model dependent. In this talk I will explain how this picture for CFTs drawn by Segal applies to Liouville theory (LCFT), which is a non rational conformal field theory developed in the early 80s in physics to describe random Riemannian metrics on Riemann surfaces. .

• April 25

Andrew Ahn (Cornell University)

Lyapunov Exponents of Random Matrix Products and Brownian Motion on GL(n,C)

Abstract: Consider the discrete-time process formed by the singular values of products of random matrices, where time corresponds to the number of matrix factors. It is known due to Oseledets' theorem that under general assumptions, the Lyapunov exponents converge as the number of matrix factors tend to infinity. In this talk, we consider random matrices with distributional invariance under right multiplication by unitary matrices, which include Ginibre matrices and truncated unitary matrices. The corresponding singular value process is Markovian with additional structure that admits study via integrable probability techniques. In this talk, I will discuss recent results on the Lyapunov exponents in the setting where the number M matrix factors tend to infinity simultaneously with matrix sizes N. When this limit is tuned so that M and N grow on the same order, the limiting Lyapunov exponents can be described in terms of Dyson Brownian motion with a special drift vector, which in turn can be linked to a matrix-valued diffusion on the complex general linear group. We find that this description is universal, under general assumptions on the spectrum of the matrix factors.

• May 2

*** Special Seminar on Zoom! ***

Massimiliano Gubinelli (University of Bonn)

What is stochastic quantization?

Abstract: In this talk I will introduce the idea of stochastic quantization from a mathematical perspective, that is as a tool to analyze rigorously Euclidean quantum fields. I will show that there are several different "stochastic quantizations” for which we will identify common structures and ideas which take the form of a stochastic analysis of Euclidean quantum fields.

• May 9

*** Special Seminar Starting at 3pm in Room 2-361! ***

Phil Sosoe (Cornell University)

Almost-optimal regularity conditions in the CLT for Wigner matrices.

Abstract: We consider linear spectral statistics for test functions of low regularity and Wigner matrices with smooth entry distribution. We show that for functions in the Sobolev space $H^{1/2 + \epsilon}$ or the space $C^{1/2 + \epsilon}$ that are supported within the spectral bulk of the semicircle distribution, the variance remains bounded asymptotically. As a consequence, these linear spectral statistics have asymptotic Gaussian fluctuations with the same variance as in the CLT for functions of higher regularity, for any $\epsilon > 0$. This result is nearly optimal in the sense that the variance does remain bounded for functions in $H^{1/2}$, and was previously known only for matrices in Gaussian Unitary Ensemble.